Finite Group Databasebeta

← #24440

Subgroups — #24440

Order 256.

969 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D42C211yesmaximal, characteristic, complemented
2C2×#469C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic, complemented
5C2×#418C211yesmaximal, characteristic, complemented
6#2414C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2011C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1870C211yesmaximal, characteristic, complemented
14#1864C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic, complemented
16C23×D4—21no—
17C23×D4C2211yescharacteristic
18C23×D4C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic, complemented
21C22×D8C2211yescharacteristic, complemented
22C22×M16C2211yescharacteristic, complemented
23D42—24no—
24D42—24no—
25C2×#127C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic, complemented
30C2×#121C2211yescharacteristic, complemented
31C2×#120—21no—
32C2×#120—21no—
33C2×C4×D4C2211yescharacteristic
34C2×C4×D4C2211yescharacteristic
35C2×#117C2211yescharacteristic, complemented
36#469—24no—
37#447—24no—
38#447—24no—
39C2×#106C2211yescharacteristic, complemented
40#418—24no—
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic, complemented
45C2×#102C2211yescharacteristic, complemented
46C2×#102C2211yescharacteristic, complemented
47C2×#102C2211yescharacteristic, complemented
48C2×#102C2211yescharacteristic, complemented
49C2×#102C2211yescharacteristic, complemented
50C2×#102C2211yescharacteristic, complemented
51C2×C22⋊C8C2211yescharacteristic, complemented
52C2×C22⋊C8C2211yescharacteristic, complemented
53C2×C22⋊C8C2211yescharacteristic, complemented
54#394—21no—
55#392—21no—
56#390—21no—
57#390C2211yescharacteristic, complemented
58#389C2211yescharacteristic, complemented
59#388—21no—
60#380C2211yescharacteristic, complemented
61#340C2211yescharacteristic, complemented
62#340C2211yescharacteristic, complemented
63#336C2211yescharacteristic, complemented
64#336C2211yescharacteristic, complemented
65C25—21no—
66C25—21no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—28no—
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—22no—
73C22×D4—22no—
74C22×D4—22no—
75C22×D4—22no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—21no—
83C22×D4—41no—
84C22×D4—41no—
85C22×D4—41no—
86C22×D4—41no—
87C22×D4—44no—
88C22×D4—44no—
89C22×D4D411yescharacteristic
90C22×D4D414yescomplemented
91C22×D4C2311yescharacteristic
92C22×D4C2311yescharacteristic
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C2×D8—22no—
99C2×D8—22no—
100C2×D8—22no—
101C2×D8—22no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—41no—
105C2×D8—41no—
106C2×D8—41no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2311yescharacteristic
114#127—22no—
115#127—22no—
116#121—28no—
117#121—22no—
118#121—22no—
119#121—22no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121—41no—
124#121D414yescomplemented
125#120—28no—
126#120—28no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4—22no—
130C4×D4—22no—
131#117—22no—
132#117—41no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4D411yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C4⋊C4C2311yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4D411yescharacteristic
149C2×C22⋊C4D411yescharacteristic
150C2×C42—21no—
151C2×C42C2311yescharacteristic
152#106—22no—
153#102—22no—
154#102—22no—
155#102—22no—
156#102—22no—
157#102—22no—
158#102—22no—
159#102—22no—
160#102—22no—
161#102—41no—
162#102D414yescomplemented
163C22⋊C8—22no—
164C22⋊C8—22no—
165C22⋊C8—41no—
166#95—21no—
167#95—21no—
168C24—21no—
169C24—24no—
170C24—21no—
171C24—21no—
172C24—24no—
173C24—21no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—42no—
182C24—81no—
183C24—81no—
184C24—81no—
185C24—81no—
186C24C2×D411yescharacteristic
187C24C2×D411yescharacteristic
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—28no—
191C2×D4—24no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—44no—
227C2×D4—44no—
228C2×D4—432no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—81no—
241C2×D4—81no—
242C2×D4—81no—
243C2×D4C2×D412yes—
244C2×D4C2×D412yes—
245C2×D4C2×D412yes—
246C2×D4C2×D412yes—
247C2×D4#4112yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D411yescharacteristic
268C22×C4C2×D412yes—
269C22×C4C2×D412yes—
270C22×C4C2×D411yescharacteristic
271C22×C4C2411yescharacteristic
272D8—44no—
273D8—44no—
274D8—44no—
275M16—44no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8C2×D412yescomplemented
288C4⋊C4—22no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—81no—
296C4⋊C4C2×D412yes—
297C4⋊C4C2×D412yescomplemented
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—22no—
301C22⋊C4—24no—
302C22⋊C4—22no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C42—21no—
310C42—21no—
311C42—42no—
312C23—21no—
313C23—22no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—21no—
318C23—21no—
319C23—22no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—22no—
329C23—48no—
330C23—48no—
331C23—48no—
332C23—48no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—84no—
357C23—84no—
358C23—84no—
359C23—84no—
360C23—84no—
361C23—84no—
362C23—84no—
363C23—84no—
364C23—84no—
365C23—84no—
366C23#12012yes—
367C23AGL(1,ℤ/8)12yes—
368C23C22×D411yescharacteristic
369D4—28no—
370D4—22no—
371D4—22no—
372D4—22no—
373D4—22no—
374D4—22no—
375D4—22no—
376D4—22no—
377D4—48no—
378D4—48no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388D4—41no—
389D4—41no—
390D4—84no—
391D4—84no—
392D4—84no—
393D4#12114yes—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—48no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4#12012yes—
426C2×C4AGL(1,ℤ/8)12yes—
427C2×C4C22×D411yescharacteristic
428C2×C4C22×D411yescharacteristic
429C2×C4C22×D411yescharacteristic
430C2×C4C2×#4111yescharacteristic
431C2×C4H(5,𝔽2)11yescharacteristic
432C2×C4H(5,𝔽2)11yescharacteristic
433C8—22no—
434C8—41no—
435C8—41no—
436C8—41no—
437C22—22no—
438C22—24no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—42no—
446C22—41no—
447C22—42no—
448C22—41no—
449C22—42no—
450C22—41no—
451C22—41no—
452C22—42no—
453C22—41no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—816no—
465C22—816no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22C2×#12011yescharacteristic
482C22D4211yescharacteristic
483C22#54611yescharacteristic
484C22C2×AGL(1,ℤ/8)11yescharacteristic
485C22C2×AGL(1,ℤ/8)11yescharacteristic
486C22#57611yescharacteristic
487C22#57611yescharacteristic
488C4—22no—
489C4—22no—
490C4—41no—
491C4—42no—
492C4—41no—
493C4—42no—
494C4—81no—
495C4—81no—
496C4C2×#12111yescharacteristic
497C4#53411yescharacteristic
498C4D4211yescharacteristic
499C4#54611yescharacteristic
500C2—42no—
501C2—42no—
502C2—44no—
503C2—82no—
504C2—82no—
505C2—161no—
506C2#241011yescharacteristic
507C2#300311yescharacteristic
508C2#300811yescharacteristic
509C2#326411yescharacteristic
510C2#327411yescharacteristic
511C2#331811yescharacteristic
512C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×D42C2semidirect
2. C2×#469C2semidirect
3. C2×#447C2semidirect
4. C2×#447C2semidirect
5. C2×#418C2semidirect
6. #2414C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2011C2semidirect
10. #1999C2semidirect
11. #1879C2semidirect
12. #1875C2semidirect
13. #1870C2semidirect
14. #1864C2semidirect
15. #1786C2semidirect
20. C22×D8C22semidirect
21. C22×D8C22semidirect
22. C22×M16C22semidirect
29. C2×#121C22semidirect
30. C2×#121C22semidirect
35. C2×#117C22semidirect
39. C2×#106C22semidirect
44. C2×#102C22semidirect
45. C2×#102C22semidirect
46. C2×#102C22semidirect
47. C2×#102C22semidirect
48. C2×#102C22semidirect
49. C2×#102C22semidirect
50. C2×#102C22semidirect
51. C2×C22⋊C8C22semidirect
52. C2×C22⋊C8C22semidirect
53. C2×C22⋊C8C22semidirect
57. #390C22semidirect
58. #389C22semidirect
60. #380C22semidirect
61. #340C22semidirect
62. #340C22semidirect
63. #336C22semidirect
64. #336C22semidirect
90. C22×D4D4semidirect
124. #121D4semidirect
162. #102D4semidirect
287. C2×C8C2×D4semidirect
297. C4⋊C4C2×D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D422
C2×#4692
C2×#4472
C2×#4182
#24142
#20412
#20342
#20112
#19992
#18792
#18752
#18702
#18642
#17862